In the Journal of Discourses, LDS apostles suggest that Jesus Himself was married, that He had children, and that He and His disciples endorsed polygamy.

It will be borne in mind that once on a time, there was a marriage in Cana of Galilee; and on a careful reading of that transaction, it will be discovered that no less a person than Jesus Christ was married on that occasion. Journal of Discourses, Volume 4, Page 259

What did the old Prophet mean when he said (speaking of Christ), “He shall see his seed, prolong his days, &c.” Did Jesus consider it necessary to fulfil every righteous command or requirement of his Father? He most certainly did. This be witnessed by submitting to baptism under the hands of John. “Thus it becometh us to fulfil all righteousness,” said he. Was it God's commandment to man, in the beginning, to multiply and replenish the earth? None can deny this, neither that it was a righteous command; for upon an obedience to this, depended the perpetuity of our race. Did Christ come to destroy the law or the Prophets, or to fulfil them? He came to fulfil. Did he multiply, and did he see his seed? Did he honor his Father's law by complying with it, or did he not? Others may do as they like, but I will not charge our Savior with neglect or transgression in this or any other duty. Journal of Discourses, Volume 4, Page 260

The grand reason of the burst of public sentiment in anathemas upon Christ and his disciples, causing his crucifixion was evidently based upon polygamy, according to the testimony of the philosophers who rose in that age. A belief in the doctrine of a
plurality of wives caused the persecution of Jesus and his followers. We might almost think they were "Mormons". Journal of Discourses, Volume 1, Page 346.

Was this ever the official teaching of the LDS Church? If so, what biblical support for these doctrines is there?

I'm going to go out on a limb on the final question and say "zero".
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kuroschFeb 12 '13 at 16:04

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As it stands the title to this question is misleading. As you can see from the answers, this was never an official teaching so implying that it was and asking if it still is a teaching, could very well give false impressions.
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ryanFeb 12 '13 at 16:54

@ryan Thanks, Ryan. I have tried to make appropriate changes.
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NarnianFeb 12 '13 at 17:35

I like how the question is phrased now. Funny, if they believed in the Trinity, I would have a wildly different answer.
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MaskedPlantFeb 12 '13 at 19:35

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I don't feel this merits its own answer, so I will use a comment. Two things A) Clearly, as pointed out, such an opinion was expressed by leaders of the church in the past B) These opinions are not considered to have ever been introduced as doctrine today by the modern LDS church. For an apologetic discussion on this exact topic, see en.fairmormon.org/Jesus_Christ/Was_Jesus_married/…. The link contains quotes regarding some biblical/mormon reasoning given for such an opinion, if you are interested.
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DougvjFeb 13 '13 at 6:00

4 Answers
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Both @MaskedPlant and @Matt spelled out how the Journal of Discourses is viewed.

I will add this; the only thing I have ever heard was during the whole "The Da Vinci Code" hoopla. And that was that the LDS church had no official opinion and that it neither endorsed or found fault with the ideas presented in that book. (Namely of course that Christ was married and had children).

So the LDS church has no official teachings on the topic, but if it were to come to light that Christ was in fact married (or even had kids!), the LDS church would have no issue with that information.

Edit:
I found the official statement made by the church on this subject from 2006:

"The belief that Christ was married has never been official Church doctrine. It is neither sanctioned nor taught by the Church. While it is true that a few Church leaders in the mid-1800s expressed their opinions on the matter, it was not then, and is not now, Church doctrine."

That was never the official teaching of the LDS Church. The Journal of Discourses is a book and is not, nor has ever been, canonical scripture, or any other type of scripture for that matter, to the LDS.

Was this ever the official teaching of the LDS Church? If so, what biblical support for these doctrines is there?

No. It is not the official teaching.

I would refer you to this answer for more information on why this has never been official teaching of the LDS church.

This is a link to lds.org with a question about the journal of Discourses. Scroll down to the question on it for the pertaining information. I copied the two sentences that I thought were the most important.

The original intent of their publication was to provide income for George D. Watt, their stenographer and publisher.

So no scriptural intent, but they were written to provide income, by someone who was not a general authority.

Though the First Presidency endorsed the publication of the Journal, there was no endorsement as to the accuracy or reliability of the contents.

So they said go ahead and publish it, but they NEVER said anything in it was canon.

As for your title question:

What does the LDS Church teach about Jesus being married, having children, and His position on polygamy?

Nothing that I know of. I have talked to some people that would say differently, but none of them could back anything up with scripture or official canon. I can find nothing on their site at this time about whether or not Jesus was married.

I'm LDS so I can tell you that there is no set doctrine on this but there always is discussions of it amongst the members. None of the answers above actually answer the second part of the question, about there being biblical proof.

True there is no reference to Christ being married in the bible directly. But what people use as proof, that it could be possible, is the first mircale Christ ever performed. Turning water into wine. Tradtionally at a wedding the groom was in charge of food and drink. For anyone to ask Christ about the lack of wine is to imply he was the groom. Of course they don't go into enough detail for it to be made clear whether he was the groom or not. Recent speculation by writers like Dan Brown has made people think that maybe Christ had a wife and children and even point to one of the few woman named in the scriptures.

The LDS members are all encrouaged to ask God ourselves for answers and to question everything we are taught and have any doctrine taught confirmed by the spirit. This is probably why the discourses weren't discouraged. The LDS church doesn't censor its members except to say to speak sparingly and cautiously about sacred things. If the church at the time didn't think the discourses were particularly sacred they would not have stopped its printing.

What I can say is if there is any evidence uncovered that he was married, the LDS would have no issue with it as we believe that marriage is a sacred ordinance and blessed by God. Marriage and children is a good thing and not a bad thing.